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Unformatted text preview: ODE, Review for Exam 1 June 9, 2008 1. Consider the autonomous equation x = x 3- x . (a) Sketch the phase line (the x axis) and the corresponding direction field on the plane. (b) Sketch graphs of some solutions. Be sure to include at least one solution with values in each interval above, below, and between critical points. (c) Suppose x ( t ) is a nonconstant solution to the above equation. If x 00 (2) = 0 what is x (2)? (c) There are two possibilities: x = 1 / 3. 2. Consider the equation x 00 + 2 x + 2 x = e- t cos 2 t . (a) Find a particular solution. (b) Find the general solution. (a) Using variation of parameters, I find a particuar solution y p ( x ) = (- cos t + 2 3 cos 3 t )( e- t cos t )+ (sin t- 2 3 sin 3 t )( e- t sin t ). (b) y = c 1 e- t cos t + c 2 e- t sin t + (- cos t + 2 3 cos 3 t )( e- t cos t ) + (sin t- 2 3 sin 3 t )( e- t sin t ). 3. (a) If e- 2 t + 2 e- t solves x 00 + cx + kx = 0 what are c, k ? (b) Same if te- t instead. (a) c = 3, k = 2 (b) c = 2, k = 1 4. Solve y 00- 4 y = e 2 t . y = c 1 e 2 t + c 2 e- 2 t + 1 4 te 2 t . 5. Consider the equation y = ty 2 ....
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